Best Known (137, 166, s)-Nets in Base 4
(137, 166, 1272)-Net over F4 — Constructive and digital
Digital (137, 166, 1272)-net over F4, using
- 41 times duplication [i] based on digital (136, 165, 1272)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (31, 45, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F64, using
- digital (91, 120, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 30, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 30, 258)-net over F256, using
- digital (31, 45, 240)-net over F4, using
- (u, u+v)-construction [i] based on
(137, 166, 1286)-Net in Base 4 — Constructive
(137, 166, 1286)-net in base 4, using
- (u, u+v)-construction [i] based on
- (36, 50, 258)-net in base 4, using
- trace code for nets [i] based on (11, 25, 129)-net in base 16, using
- base change [i] based on (6, 20, 129)-net in base 32, using
- 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- 1 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
- base change [i] based on (6, 20, 129)-net in base 32, using
- trace code for nets [i] based on (11, 25, 129)-net in base 16, using
- digital (87, 116, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- (36, 50, 258)-net in base 4, using
(137, 166, 16447)-Net over F4 — Digital
Digital (137, 166, 16447)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4166, 16447, F4, 29) (dual of [16447, 16281, 30]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4162, 16440, F4, 29) (dual of [16440, 16278, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4162, 16443, F4, 27) (dual of [16443, 16281, 28]-code), using Gilbert–Varšamov bound and bm = 4162 > Vbs−1(k−1) = 254679 951015 177619 935564 899712 291365 432296 623509 308635 651507 656212 222031 112677 660200 439023 689136 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4162, 16440, F4, 29) (dual of [16440, 16278, 30]-code), using
- construction X with Varšamov bound [i] based on
(137, 166, large)-Net in Base 4 — Upper bound on s
There is no (137, 166, large)-net in base 4, because
- 27 times m-reduction [i] would yield (137, 139, large)-net in base 4, but